By Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki

This booklet presents finished summaries of theoretical (algebraic) and computational facets of tensor ranks, maximal ranks, and average ranks, over the true quantity box. even if tensor ranks were usually argued within the complicated quantity box, it's going to be emphasised that this e-book treats genuine tensor ranks, that have direct functions in information. The booklet presents a number of fascinating principles, together with determinant polynomials, determinantal beliefs, totally nonsingular tensors, completely complete column rank tensors, and their connection to bilinear maps and Hurwitz-Radon numbers. as well as studies of how to ensure actual tensor ranks in information, international theories akin to the Jacobian approach also are reviewed in info. The publication comprises besides an available and entire advent of mathematical backgrounds, with fundamentals of confident polynomials and calculations through the use of the Groebner foundation. in addition, this ebook presents insights into numerical tools of discovering tensor ranks via simultaneous singular price decompositions.

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1. 3 Let u, n, and m be positive integers with u ≥ n and let T = (T1 ; . . ; Tm ) be a u × n × m tensor over R. Then the following conditions are equivalent. (1) T is Absolutely full column rank. (2) For any a = (a1 , . . , am ) ∈ S m−1 , m k=1 ak Tk is full column rank. ,am )∈S m−1 (the maximum of the absolute values of n-minors of m k=1 ak Tk ) > 0. m (4) For any a = (a1 , . . , am ) ∈ Rm \{0} and any b ∈ Rn \{0}, k=1 ak Tk b = 0. 2. 4 The set of u × n × m Absolutely full column rank tensors over R is an open subset of Ru×n×m .

Let K be an algebraically closed field. Let En be the n × n identity matrix and let ⎛ 0 ⎜ .. ⎜ Jn = ⎜ . ⎝0 0 ⎞ 1 0 .. ⎟ . ⎟ ⎟ · · · 0 1⎠ ··· 0 0 be an n × n superdiagonal matrix. For our convenience, we assume that J1 is the null matrix. An n × n matrix A whose elements are in K is similar to a Jordan matrix Diag(λ1 En1 + Jn1 , . . , there exists P ∈ GL(n, K) such that P−1 AP is equal to the above Jordan matrix. A diagonal element λi is an eigenvalue of A. In the case where K = C, if A∗ A = AA∗ , where A∗ is the complex conjugate transpose, then A © The Author(s) 2016 T.

Therefore, rank F ((A, X); (B, Y )) ≤ rank F (Em ; D) + rank(Y − DX ) ≤ m + (n − m) = n. 1 Let A and B be m × m matrices over F. For sufficiently large s ∈ F, (A + sDiag(1, 2, . . , m))−1 (B + sEm ) = −1 1 A + Diag(1, 2, . . , m) s 1 B + Em s has distinct eigenvalues, which are all real if F = R. 3 Let 3 ≤ m ≤ n and T = ((Em , O); A; B) ∈ TF (m, n, 3). Then, rank F (T ) ≤ m + n. 10 (Atkinson and Stephens 1979, Theorem 4; Sumi et al. rank R (n, n, 3) ≤ 2n. rank F (m, n, 3) ≤ m + n − 1. rank R (n, n, 3) ≤ 2n − 1.